Building Diversified Portfolios that Outperform Out of Sample (Hierarchical Risk Parity)

Hierarchical Risk Parity targets the instability, concentration, and out-of-sample underperformance of quadratic optimizers — Markowitz’s critical line algorithm in particular — by applying graph theory and machine-learning clustering to the covariance matrix instead of inverting it. Because no inversion is needed, HRP can allocate on an ill-conditioned or even singular covariance matrix, where a quadratic optimizer cannot run at all. The author reports lower out-of-sample variance than the critical line algorithm in Monte Carlo experiments; the “outperform out of sample” of the title refers to that simulated variance result, not to realized trading returns.

The paper is paywalled. The DOI redirects to the publisher’s login, and no openly readable version could be verified, so the practical route into the method is through its implementations rather than the text.

Two independently maintained Python libraries implement it and cite this paper directly. PyPortfolioOpt provides HRPOpt in pypfopt/hierarchical_portfolio.py, whose docstring records that the code is reproduced with permission from the author, and gives it the same API as EfficientFrontier so the two allocators are swappable (MIT). Riskfolio-Lib exposes HRP as a model of its HCPortfolio class alongside Hierarchical Equal Risk Contribution (BSD-3-Clause). Both are actively maintained and installable from PyPI.